In the second of a series of three papers on the quantum time of arrival problem [1]}, Allcock discussed the repeated projection protocol in the context of the time of arrival of a particle which is released at some point \(x<0\)  on the real line and eventually detected by a measuring device at the origin \(x=0\) . Allcock considered a sequence of measurements at discrete time intervals which project the particle to the domain \(x<0\)  till the time of detection. It was argued that this process could be effectively modeled by including an imaginary potential \(-i V_0 \theta (-x)\) , where \(\theta (x)\)  is the Heaviside function, in the free Schrodinger evolution of the particle. The strength of the potential, \(V_0\) , is related to the intervals between measurements, \(\tau \) , as \(V_0 \sim \tau ^{-1}\) . The limit \(\tau \rightarrow 0\)  gives the Zeno effect.
We are not aware of a formal derivation of the equivalence of the repeated projection dynamics with the non-Hermitian Hamiltonian [2]}, [3]}, [4]}. In more recent work [5]}, [6]}, [7]} the first detection problem using the repeated measurement scheme has been investigated in a lattice model for a quantum particle satisying the discrete Schrodinger equation. It was numerically demonstrated that a description of the repeated measurement dynamics by means of an effective non-Hermitian Hamiltonian is in fact quite accurate for small values of \(\tau \)  (more precisely for \(\tau << \gamma ^{-1}\)  where \(\gamma \)  is the transition rate between lattice sites). In an independent set of work [8]}, [9]}, for systems with discrete Hilbert spaces, the arrival time problem using the repeated measurements protocol has been treated using a renewal approach similar to what one uses for the classical Polya first passage problem. Using this approach, a number of interesting physical results on first passage in the lattice Schrodinger problem have been obtained in Refs. [10]}, [11]}, [12]}, [13]}, [14]}. In particular it has been shown that the return probability on an infinite one dimensional lattice is less than one and we do not have the Polya recurrence of the 1-D random walk. The distribution of first detection times has been shown to scale as \(1/t^3\) . Different aspects of the equivalence to the non-Hermitian Hamiltonian have been explored in Refs. [7]}, [16]}, [17]}. Other interesting results include discussions of the non-Hermitian description in many-body systems [18]}, [19]} and in the context of experiments [20]}, [21]}.
